Breakdown of the Fermi Liquid picture in one dimen

arXiv:cond-mat/9311047v223Nov1993BreakdownoftheFermiLiquidpictureinonedimensionalfermionsystems:connectionwiththeenergylevelstatisticsR.M´elin,B.Dou¸cotandP.ButaudCRTBT-CNRS,38042GrenobleBP166Xc´edexFranceAbstractUsingtheadiabaticswitchingofinteractions,weestablishaconditionfortheexistenceofelectronicquasiparticlesinaLuttingerliquid.Itinvolvesacharac-teristicinteractionstrengthproportionaltotheinversesquarerootofthesystemlength.Aninvestigationoftheexactenergylevelseparationprobabilitydistri-butionshowsthatthisinteractionscalealsocorrespondstoacross-overfromthenoninteractingbehaviourtoarathertypicalcaseforintegrablesystems,namelyanexponentialdistribution.Thelevelspacingstatisticsofaspin1/2,onebranchLuttingermodelarealsoanalyzed,aswellasthelevelstatisticsofatwocoupledchainmodel.Thefieldofstronglycorrelatedelectronsystemshasrecentlystimulatedinterestingdiscus-sionswhicharesometimeschallengingsomemoretraditionalideasonthemanybodyproblem.Forinstance,AndersonhasproposedthatthelowenergypropertiesofatwodimensionalHub-bardmodelarenotproperlydescribedbyaFermiliquidtheory[1].Inarecentpaper[2],heemphasizesthatthisquestionrequiresanonperturbativetreatment,andacarefulconsid-erationofboundaryconditions.Asaconsequenceofthedifficultyoftheproblem,alotofefforthasbeenrecentlydedicatedtonumericalinvestigationseitherwithMonteCarlometh-odsorexactdiagonalizations[3].However,theavailablesizesremainquitesmall,andtheinterpretationoftheseresultsisoftendelicate.Aratherdifferentapproachhasbeenproposed[4]recentlywiththehopetodevelopnewtoolsforextractingmoreinformationfromfinitesystemsstudies.Theseworkshaveshownthatforalargeclassoflowdimensionnalstronglycorrelatedsystems,theenergylevelsexhibitstatisticalpropertiesratherwelldescribedbyrandommatrixtheory.Forinstance,aregimeofenergylevelrepulsionisclearlyseeninmostinvestigatedcases,attheexeptionofintegrablemodelssuchasthenearestneighbororthe1/r2interactionHeisenbergspinchain.Suchabehaviorhasbeenextensivelydiscussedinthecontextofquantumchaos.Moreprecisely,ithasbeenverifiedthatmanytimereversalsym-metricalclassicalychaoticsystemsgenerateaspectrumingoodagreementwiththeGaussianOrthogonalensemblepredictions[5].Bycontrast,simpleintegrablesystemsyieldingeneraluncorrelatedenergylevelsandtheusualexponentialdistributionforenergylevelspacings[6].Inthispaper,weareinvestigatingsomepossibleconnectionsbetweensimplephysicalpropertiesofaninteractingFermisystem,suchastheexistenceoflonglivedelectronicquasi-particlesandtheenergyleveldistribution.Intuitively,iftheenergylevelsoftheinteractingsystemkeepasimpleonetoonecorrespondencewiththoseofthenoninteractingsystem,weexpectononehandFermiLiquidTheorytobevalid,andontheotherhandthestatisticalpropertiesofthespectrumtoremainqualitativelysimilarasforthefreeelectroncase.Theinterpretationofrandommatrixbehaviourisnotstraightforward.Itmaysimplyindicatethatamodelisnonintegrable.ForanormalFermiLiquid,electronicquasiparticleareexpectedonlyatlowenergiescomparedtotheFermienergy.Furthermore,alreadyfortheparticle-holephasespaces,interactionsinducenewcollectivemodes,suchasthezerosound,andtheideaofaonetoonecorrespondencewiththenoninteractinggasdoesnotholdforthewholespectrum.Clearly,itwouldbeveryinterestingtoseeifthespectrumofanormalFermiLiquidexhibitssomefeatureswhichwoulddistinguishitfromarandommatrixHamiltonian.However,thiswouldlikelyrequireanintensivenumericaleffort(sincebestcandidateswouldbeatleasttwodimensionnalsystems).Forthesakeofsimplicity,andthemotivationofdoinganalyticalcalculations,wehaveconcentratedinthisworkonaonedimensionalmodel,namelytheLut-tingermodel[7],whichisintegrableatanycouplingstrength.Interestingly,thisfeatureholds1foranysystemlength[8].Furthermore,itprovidesagoodexampleofanon-Fermiliquid,whichcanbeviewedasanontranslationinvariantfixedpointformanyinteractingsystemsinonedimension.Thispaperisorganizedasfollows.Afirstpartinvestigatestheconditionfortheexis-tenceofelectronicquasiparticles,usingtheadiabaticgenerationofeigenstates.Anexistenceconditionisestablished,fromthecombinedrequirementofhavinganegligeablegenerationofnonadiabaticcomponentsandabsenceofdecay.Thiscriterionissatisfiediftheinteractionstrengthislessthanaconstantdividedbythesquarerootofthesystemlength.Asexpected,noquasiparticlesarefoundforaninfinitesystematanyfinitevalueoftheinteractionparame-ter.ThisresultisalsorederivedfromasimpleanalysisofthesingleparticleGreen’sfunctionforafinitesystem.Thesecondpartisdevotedtothestudyofthelevelspacingdistributionastheinteractionisgraduallyincreased.Weshowthatthetypicalinteractionscalelocatingthedeparturefromthehighlydegeneratenoninteractingsystemtowardsamoregenericinte-grablemodelwithaPoissondistributionisthesameasthepreviousone.So,forthissimplesituation,noticeablechangeintheenergyleveldistributionisreflectedbythedisappearanceofelectronicquasiparticles.Then,thelasttwosectionsofthispaperarededicatedtovariantsofthismodel,namelyinthespin1/2caseandforwardscatteringonly,forbothoneandtwocoupledonedimensionnalsystems.Abriefconclusionsummarizesourresults.1Adiabaticswitchingonofinteraction